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                <li class="toctree-l2 current"><a class="reference internal current" href="./">拉普拉斯变换</a>
    <ul class="current">
    <li class="toctree-l3"><a class="reference internal" href="#_2">拉普拉斯变换</a>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#_3">拉普拉斯变换收敛域</a>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#_4">拉普拉斯逆变换</a>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#-">由零-极点图对傅里叶变换进行几何求值</a>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#_5">拉普拉斯变换的性质</a>
        <ul>
    <li class="toctree-l4"><a class="reference internal" href="#_6">线性性质：</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_7">时移性质：</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#s">s域平移</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_8">时域尺度变换</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_9">共轭</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_10">卷积</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_11">时域微分</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_12">时域积分</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_13">初值定理和终值定理</a>
    </li>
        </ul>
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    <li class="toctree-l3"><a class="reference internal" href="#_14">常用拉普拉斯变换对</a>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#_15">用拉普拉斯变换分析与表征线性时不变系统</a>
        <ul>
    <li class="toctree-l4"><a class="reference internal" href="#_16">因果性</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_17">稳定性</a>
    </li>
        </ul>
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    <li class="toctree-l3"><a class="reference internal" href="#_18">系统函数的代数属性与方框图表示</a>
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    <li class="toctree-l3"><a class="reference internal" href="#_19">单边拉普拉斯变换</a>
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                <h1 id="_1">拉普拉斯变换</h1>
<h2 id="_2">拉普拉斯变换</h2>
<p>一个单位冲激响应为<span class="arithmatex">\(h(t)\)</span>的线性时不变系统，对<span class="arithmatex">\(e^{st}\)</span>复指数输入信号的响应是</p>
<div class="arithmatex">\[
y(t)=H(s)e^{st}
\]</div>
<p>其中</p>
<div class="arithmatex">\[
H(s)=\int_{-\infty}^{\infty}h(t)e^{-st}dt
\]</div>
<p>若s为虚数（即<span class="arithmatex">\(s=j\omega\)</span>），<span class="arithmatex">\(H(s)\)</span>就对应<span class="arithmatex">\(h(t)\)</span>的傅里叶变换。对一般的复变量s来说，<span class="arithmatex">\(H(s)\)</span>就被称为<span class="arithmatex">\(h(t)\)</span>的<strong>拉普拉斯变换</strong>，其中复变量s一般可写为<span class="arithmatex">\(s=\sigma+j\omega\)</span>，拉普拉斯变换可表示为算子形式<span class="arithmatex">\(\mathcal{L}(x(t))\)</span>，变换关系可表示为<span class="arithmatex">\(x(t)\overset{\mathcal{L}}{\rightarrow}X(s)\)</span>.</p>
<p>拉普拉斯变换可以看做是输入信号乘上一个衰减/增强信号<span class="arithmatex">\(e^{-\sigma t}\)</span>后再做傅里叶变换的结果</p>
<p>不同的拉普拉斯变换可以有相同的表达式，因此需要引入<strong>收敛域（ROC）</strong>进一步确定。对于有理拉普拉斯变换（H(s)可以表达为多项式之比），s平面除了包含阴影部分的收敛域外，还有记作圈的<strong>零点</strong>和记作叉的<strong>极点</strong>，他们分别是X(s)表达式的分子零点和分母零点，这种表示称为X(s)的<strong>零-极点图</strong>。如果有重根，则需要重复零点/极点记号。</p>
<h2 id="_3">拉普拉斯变换收敛域</h2>
<blockquote>
<p>前置定义：若在某有限时间<span class="arithmatex">\(T_1\)</span>之前，x(t)＝0，则称该信号为<strong>右边信号</strong>；若在某有限时间<span class="arithmatex">\(T_2\)</span>之后，x(t)＝0，则称该信号为<strong>左边信号</strong>；一个<span class="arithmatex">\(双边信号\)</span>就是对<span class="arithmatex">\(t&gt;0\)</span>和<span class="arithmatex">\(t&lt;0\)</span>都具有无限范围的信号，可以看作左边信号和右边信号之和。</p>
</blockquote>
<p>以下给出一些可以由表达式推收敛域的性质：</p>
<ul>
<li>X(s)的收敛域在s平面内是由平行于<span class="arithmatex">\(j\omega\)</span>轴的带状区域组成的</li>
<li>对有理拉普拉斯变换来说，收敛域内不包含任何极点</li>
<li>如果x(t)是有限持续期的，并且是绝对可积的，那么收敛域就是整个s平面</li>
<li>如果x(t)是右边信号，并且<span class="arithmatex">\(Re\{s\}=\sigma_0\)</span>这条线位于收敛域内，那么<span class="arithmatex">\(Re\{s\}&gt;\sigma_0\)</span>的全部s值都一定在收敛域内</li>
<li>如果x(t)是左边信号，并且<span class="arithmatex">\(Re\{s\}=\sigma_0\)</span>这条线位于收敛域内，那么<span class="arithmatex">\(Re\{s\}&lt;\sigma_0\)</span>的全部s值都一定在收敛域内</li>
<li>如果x(t)是双边信号，并且<span class="arithmatex">\(Re\{s\}=\sigma_0\)</span>这条线位于收敛域内，那么收敛域就一定由s平面的一条带状区域组成，直线<span class="arithmatex">\(Re\{s\}=\sigma_0\)</span>位于该区域内。</li>
<li>如果x(t)的拉普拉斯变换X(s)是有理的，那么它的收敛域是被极点所界定的或延伸到无限远。另外，在收敛域内不包含X(s)的任何极点。</li>
<li>如果x(t)的拉普拉斯变换X(s)是有理的，那么若x(t)是右边信号，则其收敛域在s平面上位于最右边极点的右边；若x(t)是左边信号，则其收敛域在s平面上位于最左边极点的左边</li>
</ul>
<h2 id="_4">拉普拉斯逆变换</h2>
<div class="arithmatex">\[
x(t)=\frac{1}{2\pi j}\int_{\sigma-j\infty}^{\sigma+j\infty}X(s)e^{st}ds
\]</div>
<p>上式不好求，对于有理变换一般用类似前面的部分分式法求。</p>
<h2 id="-">由零-极点图对傅里叶变换进行几何求值</h2>
<p>简单来说，对于有理拉普拉斯变换，多项式之比中的每一项可以看作是一个s平面上从零点/极点到变量s的向量，而傅里叶变换就是拉普拉斯变换的虚轴，于是就可以根据向量模长和相位的变化对傅里叶变换定性分析。</p>
<h2 id="_5">拉普拉斯变换的性质</h2>
<p>设<span class="arithmatex">\(x(t)\overset{\mathcal{L}}{\rightarrow}X(s),ROC=R\)</span>,<span class="arithmatex">\(x_1(t)\overset{\mathcal{L}}{\rightarrow}X_1(s),ROC=R_1\)</span>,<span class="arithmatex">\(x_2(t)\overset{\mathcal{L}}{\rightarrow}X_2(s),ROC=R_2\)</span></p>
<h3 id="_6">线性性质：</h3>
<div class="arithmatex">\[
ax_1(t)+bx_2(t)\overset{\mathcal{L}}{\rightarrow}aX_1(s)+bX_2(s)
\]</div>
<p>收敛域包括<span class="arithmatex">\(R_1\cap R_2\)</span></p>
<h3 id="_7">时移性质：</h3>
<div class="arithmatex">\[
x(t-t_0)\overset{\mathcal{L}}{\rightarrow}e^{-st_0}X(s)
\]</div>
<p>收敛域为R</p>
<h3 id="s">s域平移</h3>
<div class="arithmatex">\[
e^{s_0t}x(t)\overset{\mathcal{L}}{\rightarrow}X(s-s_0)
\]</div>
<p>收敛域是R的平移，即若<span class="arithmatex">\((s-s_0)\)</span>在R中，则s就位于收敛域中</p>
<h3 id="_8">时域尺度变换</h3>
<div class="arithmatex">\[
x(\alpha t)\overset{\mathcal{L}}{\rightarrow}\frac{1}{\alpha}X(\frac{s}{\alpha})
\]</div>
<p>收敛域为<span class="arithmatex">\(R/\alpha\)</span>，即若<span class="arithmatex">\(s/\alpha\)</span>在R中，则s就位于收敛域中</p>
<h3 id="_9">共轭</h3>
<div class="arithmatex">\[
x^*(t)\overset{\mathcal{L}}{\rightarrow}X^*(s^*)
\]</div>
<p>收敛域为R</p>
<h3 id="_10">卷积</h3>
<div class="arithmatex">\[
x_1(t)*x_2(t)\overset{\mathcal{L}}{\rightarrow}X_1(s)X_2(s)
\]</div>
<p>收敛域包括<span class="arithmatex">\(R_1\cap R_2\)</span></p>
<h3 id="_11">时域微分</h3>
<div class="arithmatex">\[
\frac{d}{dt}x(t)\overset{\mathcal{L}}{\rightarrow}sX(s)
\]</div>
<p>收敛域包括R</p>
<h3 id="_12">时域积分</h3>
<div class="arithmatex">\[
\int_{-\infty}^t x(\tau)d\tau\overset{\mathcal{L}}{\rightarrow}\frac 1 s X(s)
\]</div>
<p>收敛域至少为<span class="arithmatex">\(R\cap\{Re\{s\}&gt;0\}\)</span></p>
<h3 id="_13">初值定理和终值定理</h3>
<p>若t&lt;0,x(t)=0且在t=0不包括任何冲激或高阶奇异函数，则</p>
<div class="arithmatex">\[
x(0^+)=\lim_{s\to \infty}sX(s),\lim_{t\to \infty}x(t)=\lim_{s\to0}sX(s)
\]</div>
<h2 id="_14">常用拉普拉斯变换对</h2>
<table>
<thead>
<tr>
<th>信号</th>
<th>变换</th>
<th>收敛域</th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="arithmatex">\(\delta(t)\)</span></td>
<td>1</td>
<td>全部s</td>
</tr>
<tr>
<td><span class="arithmatex">\(u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{1}{s}\)</span></td>
<td>Re{s}&gt;0</td>
</tr>
<tr>
<td><span class="arithmatex">\(-u(-t)\)</span></td>
<td><span class="arithmatex">\(\frac 1 s\)</span></td>
<td>Re{s}&lt;0</td>
</tr>
<tr>
<td><span class="arithmatex">\(\frac{t^{n-1}}{(n-1)!}u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{1}{s^n}\)</span></td>
<td>Re{s}&gt;0</td>
</tr>
<tr>
<td><span class="arithmatex">\(-\frac{t^{n-1}}{(n-1)!}u(-t)\)</span></td>
<td><span class="arithmatex">\(\frac{1}{s^n}\)</span></td>
<td>Re{s}&lt;0</td>
</tr>
<tr>
<td><span class="arithmatex">\(e^{-\alpha t}u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{1}{s+\alpha}\)</span></td>
<td><span class="arithmatex">\(Re\{s\}&gt;-\alpha\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(-e^{-\alpha t}u(-t)\)</span></td>
<td><span class="arithmatex">\(\frac{1}{s+\alpha}\)</span></td>
<td><span class="arithmatex">\(Re\{s\}&lt;-\alpha\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{1}{(s+\alpha)^n}\)</span></td>
<td><span class="arithmatex">\(Re\{s\}&gt;-\alpha\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)\)</span></td>
<td><span class="arithmatex">\(\frac{1}{(s+\alpha)^n}\)</span></td>
<td><span class="arithmatex">\(Re\{s\}&lt;-\alpha\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(\delta(t-T)\)</span></td>
<td><span class="arithmatex">\(e^{-sT}\)</span></td>
<td>全部s</td>
</tr>
<tr>
<td><span class="arithmatex">\(\cos(\omega_0t)u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{s}{s^2+\omega_0^2}\)</span></td>
<td>Re{s}&gt;0</td>
</tr>
<tr>
<td><span class="arithmatex">\(\sin(\omega_0t)u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{\omega_0}{s^2+\omega_0^2}\)</span></td>
<td>Re{s}&gt;0</td>
</tr>
<tr>
<td><span class="arithmatex">\(e^{-\alpha t}\cos(\omega_0t)u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{s+\alpha}{(s+\alpha)^2+\omega_0^2}\)</span></td>
<td><span class="arithmatex">\(Re\{s\}&gt;-\alpha\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(e^{-\alpha t}\sin(\omega_0t)u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{\omega_0}{(s+\alpha)^2+\omega_0^2}\)</span></td>
<td><span class="arithmatex">\(Re\{s\}&gt;-\alpha\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(u_n(t)=\frac{d^n(\delta(t))}{dt^n}\)</span></td>
<td><span class="arithmatex">\(s^n\)</span></td>
<td>全部s</td>
</tr>
<tr>
<td><span class="arithmatex">\(u_{-n}(t)=u(t)*\cdots*u(t),卷积n次\)</span></td>
<td><span class="arithmatex">\(\frac{1}{s^n}\)</span></td>
<td>Re{s}&gt;0</td>
</tr>
</tbody>
</table>
<h2 id="_15">用拉普拉斯变换分析与表征线性时不变系统</h2>
<p>一般将<span class="arithmatex">\(H(s)\)</span>称为系统函数或传递函数</p>
<h3 id="_16">因果性</h3>
<p>一个因果系统的系统函数的收敛域是某个右半平面。一般情况下反之不一定成立，但对于一个具有有理系统函数的系统来说，系统的因果性就等效于收敛域位于最右边极点右边的右半平面。</p>
<h3 id="_17">稳定性</h3>
<p>当且仅当系统函数<span class="arithmatex">\(H(s)\)</span>的收敛域包括<span class="arithmatex">\(j\omega\)</span>轴时，线性时不变系统时稳定的</p>
<p>当且仅当系统函数的全部极点都位于s平面的左半平面时，一个具有有理系统函数的因果系统才是稳定的</p>
<h2 id="_18">系统函数的代数属性与方框图表示</h2>
<p>有几种方法关联系统函数和方框图</p>
<ol>
<li>将系统函数拆分成多个式子相乘，根据时域卷积频域相乘，这个系统函数就可以被看成多个系统级联结果。基本的原件是1/s的积分器和乘/加节点（级联型）</li>
<li>将系统函数拆成多项式形式，根据线性性质，这个系统函数就可以被看成多个系统并联（并联型）</li>
<li>给定一个有理系统函数，由于H=Y/X，设阶数个函数，相互用积分器关联（也就是各阶导数），然后用这些函数表达Y，用含X的多项式表达其中一个函数，讲这些关系画成图（直接型）</li>
</ol>
<p>具体还是多看例题多练练</p>
<h2 id="_19">单边拉普拉斯变换</h2>
<p>单边拉普拉斯变换定义为</p>
<div class="arithmatex">\[
X(s)=\int_{0-}^\infty x(t)e^{-st}dt
\]</div>
<p>之所以引入单边拉普拉斯变换，是因为它可以用来解带初始条件的微分方程：</p>
<p>单边拉普拉斯变换的微分性质为</p>
<div class="arithmatex">\[
\frac{d}{dt}x(t)\overset{\mathcal{UL}}{\rightarrow}s\chi(s)-x(0^-)
\]</div>
<p>可以发现引入了常量<span class="arithmatex">\(x(0^-)\)</span>，如此，对具有<span class="arithmatex">\(dy(t)/dt\)</span>的差分方程两边求单边拉普拉斯变换，就会有常量<span class="arithmatex">\(y(0^-)\)</span>出现，从而可以求解带有初始条件的微分方程。</p>
              
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